Problem 20 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ 5 A $$

Step-by-Step Solution

Answer

Scalar multiplication results in \( \begin{bmatrix} 10 & -25 \\ 0 & 35 \end{bmatrix} \).

Step 1: Understand the Matrix A

Matrix \( A \) is given by \( A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \). It is a \( 2 \times 2 \) matrix.

Step 2: Understand the Operation

The operation \( 5A \) means we need to multiply each element of the matrix \( A \) by the scalar value 5.

Step 3: Perform Scalar Multiplication

To perform the multiplication, multiply each element of matrix \( A \) by 5: \[5A = 5 \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 5 \cdot 2 & 5 \cdot (-5) \ 5 \cdot 0 & 5 \cdot 7 \end{bmatrix}\] Calculating each term gives: \( 5A = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix} \).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in matrix operations. It involves multiplying every element of a matrix by a scalar, which is simply a single number. Let's break it down:

• Start with a matrix, such as matrix \( A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \).
• Given a scalar, say 5, to be multiplied with the matrix, each entry of the matrix will be multiplied by this scalar.

The operation we are performing is represented as \( 5A \). This means:

• The first element (2) becomes \( 5 \times 2 = 10 \).
• The second element (-5) becomes \( 5 \times (-5) = -25 \).
• The third element (0) stays \( 5 \times 0 = 0 \).
• The fourth element (7) turns into \( 5 \times 7 = 35 \).

Thus, the resulting matrix after the multiplication is \( \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix} \). Scalar multiplication is straightforward and allows for easy scaling of matrices in various mathematical and scientific computations.

2x2 Matrix

A 2x2 matrix is a specific type of matrix that consists of two rows and two columns. Understanding this type of matrix is crucial for many math applications, such as transformations, systems of equations, and much more.

• The general form of a 2x2 matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
• Each element in the matrix has a specific position: \( a_{11}, a_{12}, a_{21}, a_{22} \).
• The elements are typically referred to using a row-column index notation; for example, \( a_{11} \) for the element in the first row and first column.

For matrix \( A \), each entry signifies a specific value to operate upon, and this is useful for performing operations like addition, subtraction, and multiplication with other matrices or scalars. The 2x2 matrix in our exercise is \( \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \). Understanding how each of these values interacts in matrix operations is core to mastering matrix algebra.

Matrix Operations

Matrix operations encompass a wide array of activities one can perform on matrices. Let's explore some key operations briefly:

• Addition/Subtraction: Only matrices of the same dimensions can be added or subtracted. Corresponding elements are combined.
• Scalar Multiplication: As demonstrated, each element in the matrix is multiplied by a scalar.
• Matrix Multiplication: A more complex operation where the elements of row of the first matrix are multiplied with the corresponding elements of the column of the second matrix and summed up.

Each of these operations follows specific rules and properties vital for computations in fields like engineering, computer graphics, and data science. In our example, the focus was on scalar multiplication, which is a stepping stone to understanding other complex operations like matrix multiplication. Remember that each operation has its prerequisites, such as dimension compatibility, especially in case of additions and multiplication of matrices. Understanding these operations opens the door to more advanced topics and applications in linear algebra.